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A Note on Over-Smoothing for Graph Neural Networks
Chen Cai, Yusu Wang
TL;DR
Deep GNNs can lose performance through over-smoothing, while prior nonlinear analysis was limited in scope. This paper tracks Dirichlet energy to analyze general GNN architectures and shows exponential convergence under spectral and weight conditions. The approach recovers earlier bounds, handles more nonlinearities, and empirically examines edge operations.
Problem
Over-smoothing limits deep GNN performance, but prior analysis of nonlinear architectures was limited, particularly beyond ReLU.
Method
The paper analyzes embeddings by tracking their Dirichlet energy across layers and relating its decay to the augmented normalized Laplacian spectrum and weight matrices.
Results
When s¯λ < 1, the embedding Dirichlet energy satisfies E(X^(l)) ≤ O((s¯λ)^l) and converges exponentially to zero.
Takeaways & Limitations
The Dirichlet-energy proof recovers prior bounds, handles Leaky ReLU, extends to common nonlinearities on regular graphs, and shows edge operations can affect energy.
Takeaways & Limitations
The proof relies on a crucial, potentially too-strong assumption on the norm of the GNN weight function, and broader nonlinearities and architectures remain open.
Abstract
from arXiv · showhide
Graph Neural Networks (GNNs) have achieved a lot of success on graph-structured data. However, it is observed that the performance of graph neural networks does not improve as the number of layers increases. This effect, known as over-smoothing, has been analyzed mostly in linear cases. In this paper, we build upon previous results \cite{oono2019graph} to further analyze the over-smoothing effect in the general graph neural network architecture. We show when the weight matrix satisfies the conditions determined by the spectrum of augmented normalized Laplacian, the Dirichlet energy of embeddings will converge to zero, resulting in the loss of discriminative power. Using Dirichlet energy to measure "expressiveness" of embedding is conceptually clean; it leads to simpler proofs than \cite{oono2019graph} and can handle more non-linearities.
1. Introduction
GNN depth can worsen performance through over-smoothing, while prior nonlinear analyses were limited; this paper uses Dirichlet energy to extend the analysis and study edge operations.
- Problem: Deeply stacking GNN layers often produces significantly worse performance, with over-smoothing identified as one contributing phenomenon.Other associated factors include vanishing gradients and overfitting from increasing parameter counts.
- Prior analysis: Over-smoothing analyses showed that repeated Laplacian smoothing makes node features in a connected graph converge to similar values.The paper notes that the term over-smoothing is technically a misnomer because the decreasing quantity is tr(X^T ˜∆X), not real smoothness.
- Research gap: The paper extends prior expressive-power analysis of nonlinear GNNs beyond ReLU, whose treatment was previously limited to that activation.The authors identify extensions to Sigmoid and Leaky ReLU as nontrivial in prior work.
- Approach: The proposed analysis tracks Dirichlet energy of node embeddings across layers as the number of layers tends to infinity.The paper uses this energy to measure embedding expressiveness.
- Experiments: Experiments on varied graphs examine edge removal and high edge weights, finding that both operations can increase Dirichlet energy in many cases.The experiments cover common benchmarks and synthetic graphs.
2. Notation
The notation defines an augmented graph with self-loops, its normalized Laplacian and propagation matrix, and a multilayer GCN whose asymptotic output is studied.
- Graph augmentation: The graph is augmented with self-loops by defining ˜A := A + I_N and ˜D := D + I_N.A is the adjacency matrix and D is the degree matrix.
- Graph operators: The augmented normalized Laplacian is defined from the augmented adjacency and degree matrices.Its displayed expression is introduced in the notation section.
- GCN architecture: The GCN is composed of layer maps f_l(X) = MLP_l(PX), where multilayer perceptrons apply learned weight matrices and elementwise nonlinearities.Weight matrices need not be square, allowing channel dimensions to vary across layers.
- Asymptotic analysis: The embeddings are recursively defined as X^(l+1) := f_l(X^(l)), and the analysis studies X^(L) as L approaches infinity.The notation includes layer and channel sizes.
- Graph operators: The propagation matrix P satisfies P = I_N − ˜∆, with eigenvalues in (−1, 1].The eigenvalues of ˜∆ lie in [0, 2).
3. Main Result
The analysis tracks Dirichlet energy across GCN layers and shows that graph propagation, weights, and selected nonlinearities can drive it toward zero under spectral and singular-value conditions.
- Analysis strategy: Dirichlet energy is tracked across layers to measure the smoothness and expressiveness of node embeddings.The proof analyzes the effects of propagation, weight matrices, and nonlinearities separately.
- Propagation: The propagation operator satisfies E(PX) ≤ (1 − λ)^2E(X), where λ is the smallest non-zero eigenvalue of the augmented normalized Laplacian.Thus, the graph operator contracts Dirichlet energy according to the Laplacian spectrum.
- Weights: Weight transformations are controlled by the maximum singular values of the weight matrices, including when embedding dimensions vary across layers.The bound applies to rectangular matrices and depends on the product of layerwise singular-value factors.
- Nonlinearity: For ReLU and Leaky-ReLU, nonlinear activation does not increase Dirichlet energy: E(σ(X)) ≤ E(X).The argument uses their positive homogeneity; for regular graphs, it extends to Tanh and Sigmoid.
- Main theorem: E(X^(l)) ≤ O((sλ̄)^l), and Dirichlet energy converges exponentially to zero when sλ̄ < 1.Here s is the supremum of the layerwise singular-value factors and λ̄ = (1 − λ)^2.
- Scope: The conclusion shares the earlier eigengap-based analysis while covering more general weight dimensions and nonlinearities in supported settings.The paper notes direct applicability to graphs with large eigengaps.
4. Key Differences
The paper replaces the earlier distance-based perspective with Dirichlet-energy analysis, simplifying the treatment of nonlinearities and allowing varying embedding dimensions.
- Previous analysis: The earlier analysis uses d_M(X), originally defined for fixed embedding dimensions across layers.The paper modifies this framework to accommodate changing dimensions.
- Weight dimensions: The modified distance bound applies to rectangular weight matrices, extending d_M(XW) ≤ s d_M(X) beyond square matrices.The factor s is the singular value of W.
- Nonlinearities: The earlier proof is limited to ReLU because it relies on ReLU's projection and compatibility with Perron-Frobenius eigenvector properties.The authors describe extension to sigmoid and related activations as nontrivial.
- Dirichlet-energy proof: The Dirichlet-energy proof analyzes the nonlinearity term by term, making Lemma 3.3 straightforward once the energy is expanded.This supplies the paper's simpler route to handling Leaky-ReLU and other supported cases.
5. Experiments
The experiments test how edge deletion and extreme edge reweighting affect eigenvalues and Dirichlet energy on benchmark and synthetic graphs. Across nearly all graphs and tested ratios, edge deletion increases Dirichlet energy, while extreme reweighting often behaves similarly.
- Experimental setup: The experiments compare original and modified graphs after randomly dropping or reweighting 10%–90% of edges, using eigenvalues and Dirichlet energies of propagated signals.Signals include x, Px or P′x, and P^2x or P′^2x, with repeated trials producing 120 scatter-plot data points.
- Edge deletion: Dropping edges increases Dirichlet energy for nearly all graphs and ratios, except for some Cora and CiteSeer cases.This pattern holds for x, Px or P′x, and P^2x or P′^2x.
- Edge deletion: The observed increase in Dirichlet energy from edge deletion coincides with DropEdge’s reported reduction of over-smoothing.The passage frames this as an empirical correspondence rather than a new causal demonstration.
- Edge reweighting: In most cases, increasing a few edge weights from 1 to 10000 is similar to dropping many edges in terms of eigenvalues and Dirichlet energy.The paper describes these operations as dual and hypothesizes that extreme reweighting may also relieve over-smoothing.
6. Conclusion
The paper presents an alternative proof of exponentially decreasing GNN expressive power and empirically studies edge operations through Dirichlet energy. It identifies broader nonlinearities and learning-related resistance to over-smoothing as open directions.
- Conclusion: The paper provides an alternative proof that graph neural networks exponentially lose expressive power and matches the bound of Oono and Suzuki.The proof also handles Leaky ReLU.
- Conclusion: The paper empirically explores how basic edge operations affect the Dirichlet energy of graph embeddings.The experiments examine edge removal and edge-weight increases.
- Future directions: Extending the strategy to other nonlinearities, normalization strategies, graphs with node and edge features, and attention mechanisms remains largely open.The paper identifies nonlinearity as a key challenge in analyzing over-smoothing.
- Future directions: The proof’s assumption on the norm of GNN weight functions is crucial and may be too strong, motivating study of how learning resists over-smoothing.This limitation is stated as a future research direction.
A. Missing Proof
The appendix presents an alternative proof of a lemma using a generic matrix inequality. The proof expands the Dirichlet-energy expression and invokes the relationship between the largest eigenvalue and singular value.
- A. Missing Proof: The alternative proof is intended to avoid relying on any particular graph structure by using a generic matrix inequality.The passage introduces this as an alternative proof of Lemma 3.2.
- A. Missing Proof: The proof begins by expanding E(XW) in matrix form.The supplied proof passage does not state the complete expansion.
- A. Missing Proof: The argument uses λmax(A*A) = σmax(A) to relate the largest eigenvalue of A*A to A’s largest singular value.This identity is the explicit matrix fact shown in the proof materials.
B. Experiments
The experiments use synthetic graphs and the Cora and CiteSeer citation benchmarks, with lower-eigenvector thresholds adjusted to graph type. Figure 2 concerns Cora.
- B. Experiments: The experiments cover synthetic graphs and real graph benchmarks, including Cora and CiteSeer.The synthetic set includes random, random geometric, stochastic block-model, and Barabasi–Albert graphs.
- B. Experiments: The lower-eigenvector threshold T is 20 for synthetic graphs, 400 for Cora, and 600 for CiteSeer.The larger benchmark thresholds reflect their large numbers of nearly zero eigenvalues.
- Synthetic graphs: The random geometric graph has 200 uniformly random nodes in the unit cube, with edges joining nodes at distance at most 0.2.This defines the synthetic graph used in the Figure 1 caption’s setting.
- Citation benchmarks: Cora contains 2708 document nodes and 5278 citation-link edges, while CiteSeer contains 3327 nodes and 4552 citation-link edges.These are the dataset sizes reported for the two citation graphs.
- Figures: Figure 2 is identified as the Cora experiment.The supplied caption provides no additional axis, encoding, or outcome information.